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diff --git a/Master/texmf-dist/doc/amstex/siam/amsamp.tex b/Master/texmf-dist/doc/amstex/siam/amsamp.tex deleted file mode 100644 index 44048937dd7..00000000000 --- a/Master/texmf-dist/doc/amstex/siam/amsamp.tex +++ /dev/null @@ -1,365 +0,0 @@ -% This is the sample paper for the AmSTeX SIAM style file, (amstex)siam.sty -% for use with AmSTeX version 2.1 or later and amsppt.sty, version 2.1a. -% RCS information: $Revision: 1.1 $, $Date: 93/01/25 15:33:19 $. -\input amstex -\documentstyle{amstexs1} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% Macro definitions for running heads and first page % -\accepted\SIMAF % -\firstpageno{10} % -\lastpageno{12} % -\issuevolume{1} % -\issuenumber{2} % -\issuemonth{February} % -\placenumber{002} % place of paper in this issue % -\issueyear{1988} % -\shortauthor{Bradley J. Lucier and Douglas N. Arnold} % -\shorttitle{A Sample Paper} % -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% Macros specific to this paper % -\define\loner{{L^1(\Bbb R)}} % -\define\linfr{{L^\infty(\Bbb R)}} % -\define\bvr{{\roman{BV}(\Bbb R)}} % -\define\TV{{\roman {TV}}} % -\define\sdot{\,\cdot\,} % -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\topmatter -\title -A SAMPLE PAPER, WITH A RATHER LONG TITLE, TO ILLUSTRATE THE -\AmSTeX\ SIAM STYLE\footnote[\boldkey*]{Unlikely to appear.} -\endtitle -\author -BRADLEY J. LUCIER\footnote[\dag]{Department of Mathematics, Purdue University, -West Lafayette, Indiana 47907. Present address, somewhere on the beach -(lucier\@math.purdue.edu). -The work of the first author was not supported by the -Wolf Foundation.}\ and DOUGLAS N. ARNOLD\footnote[\ddag]{Department -of Mathematics, Pennsylvania State University, -University Park, Pennsylvania 16802.} -\endauthor -\abstract -This sample paper illustrates many of the amstex -macros as used with the \AmSTeX\ SIAM style file amstexsiam (version 2.0a). -The \AmSTeX\ SIAM style file, which -inputs and builds upon the amsppt style (version 2.1a or later) -of Michael Spivak, gives authors easy -access to most of the typographical constructions used in SIAM journals. -It does not address the issues of the table of contents -or tables, which must be set using more primitive \TeX\ macros. -\endabstract -\keywords -porous medium, interface curves -\endkeywords -\subjclass -65N60 -\endsubjclass -\endtopmatter -\document -\subhead 1. Introduction\endsubhead -We are concerned with numerical approximations to the so-called -porous-medium equation \cite{6}, -$$ -\alignedat2 - &u_t=\phi(u)_{xx},&&\qquad x\in\Bbb R,\quad t>0,\quad\phi(u)=u^m,\quad m>1, -\\ - &u(x,0)=u_0(x),&&\qquad x\in\Bbb R. -\endalignedat -\tag 1.1 -$$ -We assume that the initial data $u_0(x)$ has bounded support, that -$0\leq u_0\leq M$, and that $\phi(u_0)_x\in\bvr$. -It is well known that a unique solution $u(x,t)$ of (1.1) exists, -and that $u$ satisfies -$$ - 0\leq u\leq M\text{ and }\TV\phi(u(\,\cdot\,,t))_x\leq\TV\phi(u_0)_x. -\tag 1.2 -$$ -If the data has slightly more regularity, then this too is satisfied -by the solution. Specifically, if $m$ is no greater than two and -$u_0$ is Lipschitz continuous, then $u(\,\cdot\,,t)$ is also Lipschitz; -if $m$ is greater than two and $(u_0^{m-1})_x\in\linfr$, then -$(u(\,\cdot\,,t)^{m-1})_x\in\linfr$ -(see [3]). (This will follow from results presented here, also.) -We also use the fact that the solution $u$ is H\"older continuous in $t$. - -\subhead 2. $\linfr$ error bounds\endsubhead -After a simple definition, we state a theorem -that expresses the error of approximations $u^h$ in -terms of the weak truncation error $E$. -\definition{Definition 2.1}\rm A {\it definition} -is the same as a theorem set in roman -type. In version 2 of the \AmSTeX\ style file for the SIAM journals, -definitions are set with their own command. -\enddefinition -\proclaim{Theorem 2.1} -Let $\{u^h\}$ be a family of approximate solutions satisfying -the following conditions for $0\leq t\leq T${\rm:} -\roster -\item For all $x\in\Bbb R$ and positive $t$, $0\leq u^h(x,t)\leq M${\rm;} -\item Both $u$ and $u^h$ are H\"older--$\alpha$ in $x$ -for some $\alpha\in(0,1\wedge 1/(m-1))${\rm;} $u^h$ is right -continuous in $t${\rm;} -and $u^h$ is H\"older continuous in $t$ on -strips $\Bbb R\times(t^n,t^{n+1})$, with the set $\{t^n\}$ having no -limit points\/{\rm;} and -\item There exists a positive function $\omega(h,\epsilon)$ such that\/{\rm:} -whenever $\{w^\epsilon\}_{0<\epsilon\leq\epsilon_0}$ is a family of functions -in $\bold X$ for which -{\roster -\item"(a)" there is a sequence of positive numbers $\epsilon$ tending -to zero, such that for these values of -$\epsilon$, $\|w^\epsilon\|_\infty\leq 1/\epsilon$, -\item"(b)" for all positive -$\epsilon$, $\|w_x^\epsilon(\sdot,t)\|_\loner\leq 1/\epsilon^2$, and -\item"(c)" for all $\epsilon>0$, -$$ -\sup\Sb -x\in\Bbb R\\0\leq t_1,t_2\leq T\endSb -\dfrac{|w^\epsilon(x,t_2)-w^\epsilon(x,t_1)|}{|t_2-t_1|^p}\leq 1/\epsilon^2, -$$ -where $p$ is some number not exceeding $1$, -\endroster}% -then\footnote{This is an obvious ploy, but we need a footnote.} - $|E (u^h,w^\epsilon,T)|\leq\omega(h,\epsilon).$ -\item -This is the fourth item in the outer roster. -\endroster -Then, there is a constant $C=C(m,M,T)$ such that -$$\multline -\|u-u^h\|_{\infty,\Bbb R\times[0,T]}\leq C\biggl[ -\sup \biggl |\int_\Bbb R(u_0(x)-u^h(x,0)) w(x,0) \,dx\biggr|\\ -+\omega(h,\epsilon)+\epsilon^\alpha\biggr],\endmultline -\tag 2.1 -$$ -where the supremum is taken over all $w\in\bold X$. -\endproclaim - -\demo{Proof} -We assume first that $Q$ is decreasing and consider the following cases: -\case{Case\/ {\rm1:} -$b'\geq 1/2$} We have $P(1/8)\geq\delta>0$ where $\delta$ -depends only on $d$, for otherwise by (3.7) applied to $P$ and $p=\infty$, -$P$ could not attain the value $1$ at $x=1$. Similarly, for -$m=(a'+b')/2$, $Q(m)\geq\delta'>0$ for some $\delta'$ depending only on $d$ -since otherwise $Q$ cannot attain the value $1$ at $x=a'$. Hence, for -$\delta''=\min(\delta,\delta')$, -$|A(y)|\geq|m-1/8|\geq b'/4\geq\frac18\max(b',1)$ for -$y\in[0,\delta'']$. On the other hand, -$|A(y)|\leq \max(b',1)$ for all $y\in[0,1]$, so (4.2) follows for -all $1\leq p\leq\infty$. -\endcase -\case{Case\/ {\rm2:} -$b'\leq 1/2$} We have $P(3/4)\leq\delta<1$ with $\delta$ -depending only on $d$ for otherwise (3.7) applied to $1-P$ and $p=\infty$ -would show that $P$ could not attain the value $0$ at $x=0$. It follows -that $|A(y)|\geq 3/4-b'\geq 1/4$, $y\in[\delta,1]$, while $|A(y)|\leq 1$ -for all $y\in[0,1]$. Hence (4.2) follows for -all $1\leq p\leq\infty$. -\endcase -We consider now when $Q$ is increasing. We can assume that $Q$ is not -a translate of $P$, i.e\., we do not have $P(x)=Q(x+\delta)$ for some $\delta$, -for then (4.2) follows trivially. In what follows, $C$ and $\delta$ -depend on $d$, and $C$ may depend on $p$. We consider the following cases: -\case{Case\/ {\rm3:} $a'\geq 1/4$ and $b'\leq 100$} -From (3.7) for $P$ -and $p=\infty$, it follows that $P(1/8)\geq\delta$ since otherwise $P$ cannot -attain the value $1$ at $x=1$. Hence $|A(y)|\geq a'-1/8\geq1/8$ on -$[0,\delta]$. On the other hand $|A(y)|\leq b'$ for all $y\in[0,1]$ and hence -(4.2) follows for all $1\leq p\leq\infty$. -\endcase -Let $z$ be in $\bold X$. Because $E(u,\sdot,\sdot)\equiv0$, -Equation (1.5) implies that -$$ -\int_\Bbb R\Delta uz|^T_0dx=\int_0^T\int_\Bbb R -\Delta u(z_t+\phi[u,u^h]z_{xx})\,dx\,dt- -E(u^h,z,t), -\tag 2.2 -$$ -where $\Delta u=u-u^h$ and -$$ -\phi[u,u^h]=\dfrac{\phi(u)-\phi(u^h)}{u-u^h}. -$$ -Extend $\phi[u,u^h](\cdot,t)=\phi[u,u^h](\cdot,0)$ for negative $t$, and -$\phi[u,u^h](\cdot,t)=\phi[u,u^h](\cdot,T)$ -for $t>T$. -Fix a point $x_0$ and a number $\epsilon>0$. Let $j_\epsilon$ -be a smooth function of $x$ with integral $1$ and support in -$[-\epsilon,\epsilon]$, -and let $J_\delta$ be a smooth function of -$x$ and $t$ with integral $1$ and support in -$[-\delta,\delta]\times[-\delta,\delta]$; $\delta$ and $\epsilon$ are -positive numbers to be specified later. -We choose $z=z^{\epsilon\delta}$ to satisfy -$$ -\aligned - &z_t+(\delta+J_\delta*\phi[u,u^h])z_{xx}=0,\qquad x\in\Bbb R,\;0 -\leq t\leq T,\\ - &z(x,T)=j_\epsilon(x-x_0). -\endaligned -\tag 2.3 -$$ -The conclusion of the theorem now follows from (2.1) and the fact that -$$ -|j_\epsilon*\Delta u(x_0,t)-\Delta u(x_0,t)|\leq C\epsilon^\alpha, -$$ -which follows from Assumption 2. -\qquad\qed -\enddemo -\example{Example\/ {\rm 1}} This is an example of an example. -\endexample -\remark{Remark\/ {\rm 1}} Examples are set the same as definitions in -some styles, -and the same as proofs in others. What convention does this style follow? -\endremark -Sometimes you want to include a figure, as in Fig.~1. -\topinsert -\def\Bif{{\bf if\/ }}\def\Bwhile{{\bf while\/ }}\def\Belse{{\bf else\/ }} -\settabs\+\qquad&\qquad&\qquad&\qquad&\cr -\+\smc Tree Partition Algorithm \{\cr -\+&Let stack size denote the number of nodes in the\cr -\+&&subtrees stored temporarily on the local stack\cr -\+&pop I from global stack\cr -\+&set stack size := 0\cr -\+&\Bwhile (stack size $\leq$ max size and stack size + -I$\rightarrow$tree size $>$ 3 (max size)) \{\cr -\+&&process I as an interior node\cr -\+&&let min tree be the smaller of the subtrees of the two children of I\cr -\+&&let max tree be the larger of the subtrees of the two children of I\cr -\+&&\Bif (min tree$\rightarrow$tree size + stack size $>$ 3 (max size)) \{\cr -\+&&&push min tree onto the global stack\cr -\+&&\} \Belse \{\cr -\+&&&push min tree onto the local stack\cr -\+&&&set stack size := stack size + min tree$\rightarrow$tree size\cr -\+&&\}\cr -\+&&set I := max tree\cr -\+&\}\cr -\+&\Bif (I$\rightarrow$tree size + stack size $>$ 3 (max size)) \{\cr -\+&&push I onto the global stack\cr -\+&\} \Belse \{\cr -\+&&push I onto the local stack\cr -\+&\}\cr -\+&Process all subtrees on the local stack\cr -\+\}\cr -\botcaption{Fig.~1} Tree partition algorithm Tree partition algorithm -Tree partition algorithm Tree partition algorithm Tree partition algorithm -Tree partition algorithm Tree partition algorithm.\endcaption -\endinsert - -We finish with a table of all SIAM journals. -\midinsert -\topcaption{Table 1}{SIAM journal acronyms and titles}\endcaption -\settabs\+\indent&Acronym\indent&Title&\cr -\hbox to \hsize{\hrulefill} -\+&Acronym&Title&\cr -\hbox to \hsize{\hrulefill} -\+&SINUM&SIAM Journal on Numerical Analysis&\cr -\+&SIREV&SIAM Review&\cr -\+&SIMA&SIAM Journal on Mathematical Analysis&\cr -\+&SIMAX&SIAM Journal on Matrix Analysis and Applications&\cr -\+&SICOMP&SIAM Journal on Computing&\cr -\+&SISC&SIAM Journal on Scientific Computing&\cr -\+&SIOPT&SIAM Journal on Optimization&\cr -\+&SIAP&SIAM Journal on Applied Mathematics&\cr -\+&SICON&SIAM Journal on Control and Optimization&\cr -\+&SIDMA&SIAM Journal on Discrete Mathematics&\cr -\+&TVP&Theory of Probability and Its Applications&\cr -\hbox to \hsize{\hrulefill} -\endinsert - -\Refs -\ref - \no 1 - \by L. A. Caffarelli and A. Friedman - \paper Regularity of the free boundary of a gas flow in an - $n$-dimensional porous medium - \jour Indiana Math. J. - \vol 29 - \yr 1980 - \pages 361--391 -\endref -\ref\no 2 - \by R. DeVore and B. Lucier - \paper High order regularity for solutions of the inviscid Burgers equation - \inbook Nonlinear Hyperbolic Problems -\procinfo Proceedings of an Advanced Research Workshop, Bordeaux, -France, June 1988 - \bookinfo Lecture Notes in Mathematics - \vol 1402 - \eds C. Carasso, P. Charrier, B. Hanouzet, and J.-L. Joly - \yr 1989 - \publ Springer-Verlag - \publaddr New York - \pages 147--154 -\endref -\ref \no 3 - \bysame - \paper Wavelets - \jour Acta Numerica - \yr 1992 - \ed A. Iserles - \publ Cambridge University Press - \publaddr New York - \pages 1--56 -\endref -\ref \no 4 - \by R. A. DeVore and V. A. Popov - \paper Interpolation spaces and non-linear approximation - \inbook Function Spaces and Applications - \bookinfo Lecture Notes in Mathematics - \procinfo Proceedings of the US--Swedish Seminar held in Lund, -Sweden, June 15--21, 1986 - \vol 1302 - \eds M. Cwikel, J. Peetre, Y. Sagher, and H. Wallin - \publ Springer-Verlag - \publaddr New York - \yr 1988 - \pages 191--205 - \endref -\ref \no 5 - \by R. A. DeVore and X. M. Yu - \paper Nonlinear $n$-widths in Besov spaces - \inbook Approximation Theory VI: Vol. 1 - \eds C. K. Chui, L. L. Schumaker, and J. D. Ward - \publ Academic Press - \publaddr New York - \yr 1989 - \pages 203--206 - \lang In Russian - \endref -\ref - \no 6 - \by K. Hollig and M. Pilant - \paper Regularity of the free boundary for the porous medium equation - \paperinfo MRC Tech. Rep. 2742 -\endref -\ref - \no 7 - \by J. Jerome - \book Approximation of Nonlinear Evolution Systems - \publ Academic Press - \publaddr New York - \yr 1983 -\endref -\ref - \no 8 - \manyby R. J. LeVeque - \paper Convergence of a large time step generalization of Godunov's method - for conservation laws - \jour Comm. Pure Appl. Math. - \vol 37 - \yr 1984 - \pages 463--478 -\endref -\ref\no 9 - \by O. Rioul and M. Vetterli - \paper Wavelets and signal processing - \jour IEEE Signal Processing Magazine - \vol 8 - \issue 4 - \yr 1991 - \toappear -\endref -\endRefs -\enddocument - |